# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### How to find the smallest set of generating elements in a group?

Is there a systematic procedure for finding the smallest set of generating elements of a finite group?
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### Relation between permutation group and algebraic equation. [duplicate]

What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...
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### Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
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### How string isomorphism is used in graph isomorphism?

Graph isomorphism is a special case of string isomorphism problem. In the paper of Graph Isomorphism in Quasipolynomial Time, the relation has been shown. Let, two strings $x,y$ are associated with ...
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### Prove that L(x) = (x - 1)/p is a discrete logarithm function in a group.

I have the following problem: Let $p$ be a prime number and $G$ be a set of all $x\in \mathbb{Z}_{p^2}$, such that $x \equiv 1 \pmod{p}$. Prove that: $G$ is a multiplicative group (regarding ...
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### String isomorphism definition: Is it for any arbitrary group?

Scott Aaronson's blog, I find the description of string isomorphism as- you’re given two strings $x$ and $y$ over some finite alphabet, as well as the generators of a group $G$ of permutations ...
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### Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
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### Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....
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### Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
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### Infinite groups with all elements of order 2?

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite. This question can be proved false if we can get a group of infinite order with ...
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### Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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### Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
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### Irreducible representation of $1$-transposition groups

I would like to know the theory of irreducible representation of $1$-transposition groups. Could anyone provide me a pointer from where I can proceed?
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### Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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### Automorphism groups of partially cycle graphs

I define partially cycle graphs as follows. If we add the same subgraph to $n-k$ vertices of an $n$-vertex cycle graph, where $1\le k < n$, we create a partially cycle graph. Here are a few ...
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### Groups and rings of order $p^2$.

Up to isomorphism there are exactly two abelian groups of order $p^2$. there are exactly two groups of order $p^2$. there are exactly two commutative rings of order $p^2$. there is exactly one ...
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### Let $G = \mathbb{Z}_5 \times A_5$. if $H$ be a subgroup of $G$ of order $5$, then $H$ is weakly $s$-permutably embedded in $G$.

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some $s$-...
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### If a finite group G is a normal subgroup of the automorphism group of any connected Cayley graph thereof, is it realizable over Q?

The question is in the title: let $G$ be a finite group, and assume it is a normal subgroup of the automorphism group of any of its connected Cayley graphs. Does this imply that $G$ is the Galois ...
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### How many groups of order $2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2\cdot 13^2$ exist?

The calculation of the number of groups of order $$2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2$$ (result $81883$) takes already two hours with GAP. So, the calculation of the number of groups of order ...
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### Is a formula for $gnu(2pq^2)$ known, where $q=2p+1\$?

Let $p$ be an odd prime such that $q:=2p+1$ is also prime. Denote $g(p):=gnu(2pq^2)$ = number of groups of order $2pq^2$ upto isomorphy. The following table shows the first few values ...
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### If a conjugacy class intersect with its centralizer what can be said about its elements?

Suppose that $G$ is a finite group and let $x\in G$. If $y\in x^{G}\cap C_{G}(x)$, what can be said about the relationship of $x$ and $y$, or anything about $x$?